Quiz Chapter 08: Gauss-Seidel Method

MULTIPLE CHOICE TEST

(All Tests)

GAUSS-SEIDEL METHOD

(More on Gauss-Seidel Method)


Pick the most appropriate answer


1. A square matrix \left[ A \right]_{n \times n} is diagonally dominant if

 
 
 
 

2. Using \begin{bmatrix} x_{1}&x_{2}&x_{3} \\ \end{bmatrix} = \begin{bmatrix} 1&3&5 \\ \end{bmatrix} as the initial guess, the value of \begin{bmatrix} x_{1}&x_{2}&x_{3} \\ \end{bmatrix} after three iterations of Gauss-Seidal method is

      • \begin{bmatrix} 12&7&3 \\ 1&5&1 \\ 2&7&-11 \\ \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} = \begin{bmatrix} 2 \\ -5 \\ 6 \\ \end{bmatrix}
 
 
 
 

3. To ensure that the following system of equations,

      • 2x_{1} + 7x_{2} - 11x_{3} = 6
      • x_{1} + 2x_{2} + x_{3} = -5
      • 7x_{1} + 5x_{2} + 2x_{3} = 17

converges using the Gauss Seidal method, one can rewrite the above equations as follows:

 
 
 
 

4. For \begin{bmatrix} 12&7&3 \\ 1&5&1 \\ 2&7&-11 \\ \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} = \begin{bmatrix} 22 \\ 7 \\ -2 \\ \end{bmatrix} and using \begin{bmatrix} x_{1}&x_{2}&x_{3} \\ \end{bmatrix} = \begin{bmatrix} 1&2&1 \\ \end{bmatrix} as the initial guess, the value of \begin{bmatrix} x_{1}&x_{2}&x_{3} \\ \end{bmatrix} are found at the end of each iteration as

Iteration # x_{1} x_{2} x_{3}
1 0.41666 1.1166 0.96818
2 0.93989 1.0183 1.0007
3 0.98908 1.0020 0.99930
4 0.99898 1.0003 1.0000

At what first iteration number would you trust at least 1 significant digit in your solution?

 
 
 
 

5. The algorithm for the Gauss-Seidal method to solve \left[ A \right] \left[ X \right] = \left[ C \right] is given as follows when using nmax iterations. The initial value of [X] is stored in [X].

 
 
 
 

6. Thermistors measure temperature, have a nonlinear output and are valued for a limited range. So when a thermistor is manufactured, the manufacturer supplies a resistance vs. temperature curve. An accurate representation of the curve is generally given by

      • \dfrac{1}{T} = a_{0} + a_{1} \ln \left( R \right) + a_{2} \left[ \ln \left( R \right) \right]^{2} + a_{3} \left[ \ln \left( R \right) \right]^{3}

Where T is temperature in Kelvin, R is resistance in ohms, and a_{0},a_{1},a_{2},a_{3} are constants of the calibration curve. Given the following for a thermistor

R (ohms) T \left( ^{\circ} C \right)
1101.0 25.113
911.3 30.131
636.0 40.120
451.1 50.128

the value of the temperature in ^{\circ}C for a measured resistance of 900 ohms most nearly is